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Abstract:
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We consider a semiparametric generalized linear model and study estimation of both marginal and quantile effects in this model, which are receiving more attention due to the recent interest in mean treatment effect or outcome estimation. We propose a B-spline based approximate maximum likelihood estimator, and rigorously establish the consistency, the asymptotic normality, and the semiparametric efficiency of our method. Simulation studies are conducted to illustrate the finite sample performance, and we apply the new tool to analyze a Swiss non-labor income data and discover a new interesting predictor. The main contribution of the manuscript are the following. First, we discover that kernel estimation may not be a good approach to estimate the functional component, while B-spline estimation fits naturally and has a clear advantage. Second, we provide a clear and transparent estimation procedure, which is computationally simple and enjoys the advantage of convex optimization. Third, we establish the optimality of the resulting marginal and quantile effect estimators as functionals of the model parameters.
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